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Research of log-hyponormal operator and the Furuta inequality

Research Project

Project/Area Number 12640187
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTohoku Pharmaceutical University

Principal Investigator

TANAHASHI Kotaro  Tohoku Pharmaceutical University, Assistant Professor, 薬学部, 助教授 (90142398)

Co-Investigator(Kenkyū-buntansha) MIURA Yasuhiko  Iwate University, Faculty of Humanities and Social Sciences, Professor, 人文社会科学部, 教授 (20091647)
TAKEMOTO Hideo  Miyagi University of Education, Department of Mathematics, Professor, 教育学部, 教授 (00004408)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
KeywordsFuruta inequality / p-hyponormal operator / log-hyponormal operator / p-quashyponormal operator / log-hyponormal operator / log-hyponormal / p-hyponormal / Operator inequality
Research Abstract

T. Furuta discovered a very interesting operator inequality, which is an extension of Lowner-Heinz's inequality. Now the inequality is called the Furuta inequality and many generalizations and application has been developed. T. Furuta proved the grand Furuta inequality which contains the Furuta inequality and Ando-Hiai's inequality. Also, M. Fujii proved many inequalities with respect to chaotic order. A. Aluthge used the Furuta inequality to study p-hyponormal operators and succeeded to prove many interesting properties of p-hyponormal operators. The aim of this research is to develop the theory of operator inequality and study the properties of p-hyponormal, log-hyponormal and related classes of operators.
In this research, Tanahashi proved the best possibility of grand Furuta inequality and proved that the Furuta inequality holds for Banach ^*-algebra with A. Uchiyama. Tanahashi proved Putnam's inequality and angular cutting property holds for log-hyponormal operators. Also, Tanahash … More i proved that the Riesz idempotent for non-zero isolated point of spectrum of p-hyponormal, p-quasihyponormal, log-hyponormal, class A operators is self-adjoint with M. Cho, A. Uchiyama. Also, Tanahashi proved Schwarz type operator inequalities which are extensions of Heinz-Kato inequality with A. Uchiyama and M. Uchiyama. Also, Tanahashi proved many spectral relations of Aluthge transform with M. Cho.
Takemoto introduced a notion of numerical ranges for the elements of any von Neumann algebra by using the predual space of von Neumann algebra and showed the following properties : The first result is that the introduced notion of numerical range for any element of von Neumann algebra acting on a Hilbert space equivalents to the usual notion of numerical ranges for any operator on a Hilbert space. The second result is that by using the aboved mentioned result the numerical range of each operator is invariant under the ^*-isomorphism of the von Neumann algebra generated by the element.
Miura introduced the partial order on the set of all bounded operators on a complex Hilbert space with a selfdual cone in the sense that the difference of two operators preserves the cone and showed the Radon-Nikodym type theorem for operators a standard Hilbert space. Moreover, Miura proved the majorization property of operators of Hilbert-Schmidt class in a matrix ordered standard form from the point of view of complete positivity. Miura also proved that a not necessarily ^*-preserving homomorphism between matrix ordered von Neumann algebras and a complete order homomorphism between the underlying Hilbert spaces correspond to each other by applying the characterization of non-commutative L2-spaces in the work of Schmitt-Wittstock. In particular, a ^*-preserving homomorphism corresponds to a orthogonal decomposition homomorphism. Less

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report

Research Products

(73 results)

All Other

All Publications

  • [Publications] K.Tanahashi: "The best possibility of the grand Furuta inequality"Proceedinds of the American Mathematical Society. 128. 511-519 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "The Furuta inequality in Banach *-algebras"Proceedinds of American Mathematical Society. 128. 1691-1695 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Ishikawa, Y.Miura: "Matrix inequalities associated with a selfdual cone"Far East J.Math.Sci.(FJMS). 2. 425-431 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Ishikawa, Y.Miura: "自己共役錐体に付随する作用素の不等式について"京都大学数理解析研究所講究録. 1144. 31-38 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Ishikawa, Y.Miura: "Complete order isomorphisms between non-commutative L2-spaces"Math.Scand.. 87. 64-72 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Takemoto, N.Kakuta: "A remark on the numerical ranges of operators on Hilbert spaces"Bull.Miyagi University of Education. 35. 71-76 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, M.Cho, A.Uchiyama: "P-hyponormal, p-quasihyponormal作用素のスペクトラムの孤立点"京都大学数理解析研究所講究録. 1189. 66-81 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, K.Tanahashi: "Angular cutting for log-hyponormal operators"Integral Equations and Operator Theory. 40. 383-390 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, I.Jeon, I.Jung, J.Lee, K.Tanahashi: "Joint spectra of n-tuples of generalized Aluthge transformations"Revue de Roumaine Math.Pures et Appl.. 46. 725-730 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Miura, K.Nishiyama: "Complete orthogonal decomposition homomorphisms between matrix ordered Hilbert spaces"Proceedings of the American Mathematical Society. 129. 1137-1141 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "An example of non-selfadjoint Riesz idempotent for hyponormal operator"東北薬科大学一般教育関係論集. 15. 51-61 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, A.Uchiyama: "Isolated point of spectrum of p-quasihyponormal operators"Linear Algebra and its applications. 341. 345-350 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, A.Uchiyama, M.Uchiyama: "On the Heinz-Kato-Furuta inequality"Proc.Kotac International conference Operator Theory and Its Applications. 4. 129-134 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, K.Tanahashi: "Spectral relations for Aluthge transform"Scientiac Mathematicae Japonicae. 55. 113-119 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, K.Tanahashi: "Isolated point of spectrum of p-hyponormal, log-hyponormal operators"Integral Equations and Operator Theory. 43. 379-384 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, A.Uchiyama, M.Uchiyama: "Notes on the Heinz-Kato-Furuta-Uchiyama inequality"京都大学数理解析研究所講究録. 1259. 79-86 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Uchiyama, K.Tanahashi: "On the Riesz idempotent of class A operators"Mathematical Inequalities and Applications. 5. 291-298 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Uchiyama, K.Tanahashi: "Fuglede-Putnam's theorem for p-hyponormal or log-hyponormal operators"Glasgow Mathematical Journal. 44. 397-410 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Miura: "Remarks on selfdual cones"京都大学数理解析研究所講究録. 1259. 150-164 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Takemoto, A.Uchiyama: "A remark on the numerical ranges of operators on Hilbert spaces"Nihonkai Mathematical Journal. 13. 1-7 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "Tensor product of log-hyponormal operators"北海道大学数学講究録. 73. 32-35 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "Putnam's inequality for log-hyponormal operators"Integral Equations and Operator Theory. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, M.Uchiyama, A.Uchiyama: "On Schwarz type inequalities"Proceedings of the American Mathematical Society. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Miura: "On order of operators preserving selfdual cones in standard forms"Far East J.Math.Sci. (FJMS). (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "The best possibility of the grand Furuta inequality"Proc.Amer.Math.Soc.. 128. 511-519 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "The Furuta inequality in Banach *-algebras"Proc.Amer.Math.Soc.. 128. 1691-1695 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Ishikawa and Y.Miura: "Matrix inequalities associated with a selfdual cone"Far East J.Math.Sci. (FJMS). 2. 425-431 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Ishikawa and Y.Miura: "Operator inequalities associated with a selfdual cone"RIMS KOKYUROKU. 1144. 31-38 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Ishikawa and Y.Miura: "Complete order isomorphisms between non-commutative L2-spaces"Math.Scand.. 87. 64-72 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, M.Cho, A.Uchiyama: "Isolated point of spectrum of p-hyponormal, p-quasihyponormal operators"RIMS Kokyuroku. 1189. 66-81 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, K.Tanahashi: "Angular cutting for log-hyponormal operators"Int.Equ.Oper.Theory. 40. 383-390 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, I.Jeon, I.Jung, J.Lee, K.Tanahashi: "Joint spectra of n-tuples of generalized Aluthge transformations"Revue de Roumaine Math. Pures et Appl.. 46. 725-730 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Miura and K.Nishiyama: "Complete orthogonal decomposition homomorphisms between matrix ordered Hilbert spaces"Proc.Amer.Math.Soc.. 129. 1137-1141 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "An example of non-selfadjoint Riesz idempotent for hyponormal operator"Tohoku Pharmaceutical Univ. Studies in Liberal. 15. 51-61 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, A.Uchiyama: "Isolated point of spectrum of p-quasihyponormal operators"Linear Algebra and its Appl.. 341. 345-350 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, A.Uchiyama, M.Uchiyama: "On the Heinz-Kato-Furuta"Proc. Kotac Inter. Conf. Operator Theory and Its Applications. 4. 129-134 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, K.Tanahashi: "Spectral relations for Aluthge transform"Sci.Math.Japonicae. 55. 113-119 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, K.Tanahashi: "Isolated point of spectrum of p-hyponormal, log-hyponormal operators"Integral Equ. Oper. Theory. 43. 379-384 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, A.Uchiyama, M.Uchiyama: "Notes on the Heinz-Kato-Furuta-Uchiyama inequality"RIMS Kokyuroku. 1259. 79-86 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Uchiyama, K.Tanahashi: "On the Riesz idempotent of class A operators"Math.Ineq.Appl.. 5. 291-298 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Uchiyama, K.Tanahashi: "Fuglede-Putnam's theorem for p-hyponormal or log-hyponormal operators"Glasgow Math. J.. 44. 397-410 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Miura: "Remarks on selfdual cones"RIMS Kokyuroku. 1259. 150-164 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Takemoto and A.Uchiyama: "A remark on the numerical ranges of operators on Hilbert spaces"Nihonkai Math.J.. 13. 1-7 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "Tensor product of log-hyponormal operators"Hokkaido Univ. Technical Report Series in Mathematics. 73. 32-35 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi: "Putnam's inequality for log-hyponormal operators"Integral Equ. Oper. Theory. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Tanahashi, M.Uchiyama, A.Uchiyama: "On Schwarz type inequalities"Proc.Amer.Math.Soc.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Miura: "On order of operators preserving selfdual cones in standard forms"Far East J.Math.Sci. (FJMS). to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Cho, I.Jeon, I.Jung, J.Lee, K.Tanahashi: "Joint spectra of n-tuples of generalized Aluthge transformations"Revue de Roumanie Math. Pures et Applications. 46. 725-730 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Tanahashi: "An example of non-selfadjoint Risez idempotent for hyponormal operator"東北薬科大学一般教育関係論集. 15. 51-61 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Tanahashi, A.Uchiyama, M.Uchiyama: "On the Heinz-Kato-Furuta inequality"Proc. Kotac International Conference Operator Theory and Its Applications. 4. 129-134 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.cho, K.Tanahashi: "Isolated point of spectrum of p-hyponormal, log-hyponormal operators"Integral Equations and Operator Theory. 43. 379-384 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Tanahashi, A.Uchiyama, M.Uchiyama: "Notes on the Heinz-Kato-Furuta-Uchiyama inequality"京都大学数理解析研究所講究録. 1259. 79-86 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Miura: "Remarks on selfdual cones"京都大学数理解析研究所講究録. 1259. 150-164 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.takemoto, A.Uchiyama: "A remark on the numerical ranges of operators on Hilbert spaces"Nihonkai Mathematical Journal. 13. 1-7 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Uchiyama, K.Tanahashi: "On the Risez idempotent of class A operators"Mathematical Inequalities and Applications. 5. 291-298 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Uchiyama, K.Tanahashi: "Fuglede-Putnam's theorem for p-hyponormal or log-hyponormal operators"Glasgow Mathematical Journal. 44. 397-410 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Tanahashi: "Tensor product of log-hyponormal operators"北海道大学数学講究録. 73. 32-35 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Miura: "On order of operators preserving selfdual cones in standard forms"Far East J. Math. Sci. (FJMS). (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Tanahashi, M.Uchiyama: "On Schwarz type inequalities"Proceedings of American Mathematical Society. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Tanahashi: "Putnam's inequality for log-hyponormal operators"Integral Equations and Operator Theory. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Cho, K.Tanahashi: "Angular cutting for log-hyponormal operators"Integral Equations and Operator Theory. 40. 383-390 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Tanahashi, A.Uchiyama: "Isolated point of spectrumof p-quasihyponormal operators"Linear Algebra and its applications. 341. 345-350 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Cho, K.Tanahashi: "Spectral relations for Aluthge transform"Scientiae Mathematicae Japonicae. 55. 113-119 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Takemoto, A.Uchiyaina: "A remark of the numerical ranges of operators on Hilbert spaces"Nihonkai Matheniatical Journal. 13(to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Miura, K.Nishiyama: "Complete orthogonal decomposition homomorphisms between matrix ordered Hilbert spaces"Proceedings of American Mathematical Society. 129. 1137-1141 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kotaro Tanahashi: "The best possibility of the grand Furuta inequality"Proceedings of American Mathematical Society. 128(2). 511-519 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kotaro Tanahashi: "The Furuta inequality in Banach *-algebras"Proceedings of American Mathematical Society. 128(6). 1691-1695 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Cho and K.Tanahashi: "Putnam's inequality for log-hyponormal operators"Integral Equations and Operator Theory. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Cho and K.Tanahashi: "Angular cutting of log-hyponormal operators"Integral Equations and Operator Theory. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Miura: "Complete order isomorphisms between non-commutative L^2 spaces"Math.Scand.. 87. 64-72 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Ishikawa,Y.Miura: "Matrix inequalities associated with a self dual cone"Far East J.Math.Sci.. 2. 425-431 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Miura,K.Nishiyama: "Complete orthogonal decomposition homomorphisms between matrix ordered Hillert spaces"Proceedings of American Mathematical Society. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Takemoto,N.Kakuta: "A remark of the numerical ranges of operators"Bull.Miyagi University of Education. (to appear).

    • Related Report
      2000 Annual Research Report

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Published: 2000-03-31   Modified: 2016-04-21  

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